\(\frac{3+\sqrt{5}}{2\sqrt{2}+\sqrt{3+\sqrt{5}}}+\frac{3-\sqrt{5}}{2\sqrt{2}-\sqrt{3-\sqrt{5}}}=\frac{\sqrt{2}\left(3+\sqrt{5}\right)}{\sqrt{2}\left(2\sqrt{2}+\sqrt{3+\sqrt{5}}\right)}+\frac{\sqrt{2}\left(3-\sqrt{5}\right)}{\sqrt{2}\left(2\sqrt{2}-\sqrt{3-\sqrt{5}}\right)}=\frac{3\sqrt{2}+\sqrt{10}}{4+\sqrt{6+2\sqrt{5}}}+\frac{3\sqrt{2}-\sqrt{10}}{4-\sqrt{6-2\sqrt{5}}}=\frac{3\sqrt{2}+\sqrt{10}}{4+\sqrt{5+2\sqrt{5}+1}}+\frac{3\sqrt{2}-\sqrt{10}}{4-\sqrt{5-2\sqrt{5}+1}}=\frac{3\sqrt{2}+\sqrt{10}}{4+\sqrt{\left(\sqrt{5}+1\right)^2}}+\frac{3\sqrt{2}-\sqrt{10}}{4-\sqrt{\left(\sqrt{5}-1\right)^2}}=\frac{3\sqrt{2}+\sqrt{10}}{4+\sqrt{5}+1}+\frac{3\sqrt{2}-\sqrt{10}}{4-\sqrt{5}+1}=\frac{3\sqrt{2}+\sqrt{10}}{5+\sqrt{5}}+\frac{3\sqrt{2}-\sqrt{10}}{5-\sqrt{5}}=\frac{\left(3\sqrt{2}+\sqrt{10}\right)\left(\sqrt{5}-1\right)}{\sqrt{5}\left(\sqrt{5}+1\right)\left(\sqrt{5}-1\right)}+\frac{\left(3\sqrt{2}-\sqrt{10}\right)\left(\sqrt{5}+1\right)}{\sqrt{5}\left(\sqrt{5}+1\right)\left(\sqrt{5}-1\right)}=\frac{3\sqrt{10}-3\sqrt{2}+5\sqrt{2}-\sqrt{10}+3\sqrt{10}+3\sqrt{2}-5\sqrt{2}-\sqrt{10}}{\sqrt{5}\left(\sqrt{5}+1\right)\left(\sqrt{5}-1\right)}=\frac{4\sqrt{10}}{\sqrt{5}\left(\sqrt{5}+1\right)\left(\sqrt{5}-1\right)}=\frac{4\sqrt{2}}{5-1}=\frac{4\sqrt{2}}{4}=\sqrt{2}\)
